Since 2005 a class of algebras, the {\it Leavitt path algebras} $L_K(E)$ (for $K$ any field and $E$ any directed graph), has been a focus of investigation by both algebraists and C$^*$-analysts. In this talk I'll define these algebras, and give some of their general properties. Then I'll describe some of the current lines of investigation in the area. In particular, I'll show a connection between ideas from symbolic dynamics (``flow equivalence") and the Grothendieck group $K_0(L_K(E))$. With that connection in mind, I'll explain one of the most compelling open problems in Leavitt path algebras, the {\it Algebraic Kirchberg Phillips Question}, which can be paraphrased as: can we recover $L_K(E)$ from $K_0(L_K(E))$? While the answer to the corresponding question for graph C$^*$-algebras is {\it Yes}, there remains a barrier to a complete answer on the algebra side.